There a several ways to finding a jordan basis of a matrix $A$. For this question I have in mind two methods. Let $N=A-I\lambda$. Say we have already discovered several strings, say these two $v_1,Nv_1,...,N^3v_1$ and $v_2,Nv_2,...,N^3v_2$. Now we need a third string say length $4$. We can either (Method 1) start with $v_3$ and multiply by $A$ etc, or (Method 2) start with some $N^4w=v_3$ and solve for $w$ etc. My question is, in either case, will it be necessary to ensure linearly independence of the vectors that you find in the new string compared to the old strings? Moreover, are there any conditions that help you avoid this check?
A related question a posted earlier: Under what conditions are $v_1,Tv_1,...,T^{m_1-1}v_1,...,v_n,Tv_n,...,T^{m_n-1}v_n$ linearly independent?